First slide

Derivatives

Question

$Suppose a differntiable function f (x) satisfies the identity$
$f (x + y) = f (x) + f (y) + x y^{2} + x^{2} y_{\pm}, for all real x and y . If \lim_{x \to 0} \frac{f (x)}{x} = 1, then f^{'} (3) is$ equal to ___________ .

Moderate

Solution

From first principle of differentiation
$\begin{array}{l} f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} = \\ = \lim_{h \to 0} \frac{f (x) + f (h) + x h^{2} + x^{2} h - f (x)}{h} \\ = \lim_{h \to 0} \frac{f (h)}{h} + x^{2} \\ = 1 + x^{2} \end{array}$
$T h e r e f o r e,$
$f' (3) = 3^{2} + 1 = 10$

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